In quantum mechanics, particles are described by their wave function. Wave functions keep evolving in a continuous, predictable manner until a measurement is made on the particle, and then the wave function "collapses", meaning the particle goes into a definite state. For instance, normally quantum particles don't have a definite position, but if you measure their position their wave function suddenly collapses and they acquire a definite position.
What's so special about entanglement is that instead of each particle possessing its own wave function, the system of the two particles are described by one big wave function. When either of the particles is measured, the wave function of the whole system collapses, and thus both particles go into definite states, even if they were separated by thousands of miles. So if you have two photons that are entangled, and you measure the polarization of one of them at a certain angle, and at the same time you measure the polarization of the other one at the same angle you'll find that they're both the same, however far apart they happen to be.
It should be noted, however, that the same quantum theory which tells us about this instantaneous connection between particles also tells us that it is impossible for people to use this connection to send information faster than light. That's called the no-communication theorem.
It should also be noted that many people's first reaction, including Einstein himself, when encountering this phenomenon is to assume that the quantum mechanical explanation is wrong or incomplete, and that when the two particles were created they were assigned a common piece of information, called a "hidden variable", which they are using to coordinate their behavior. There are, however, good reasons to believe that this kind of explanation of quantum entanglement doesn't work. See "quantumtantra.com/bell2.html" for a good explanation of this.
